272    233 


ANSIT^LEVEL 


LAW 


IN  MEMORIAM 
FLORIAN  CAJORI 


ADJUSTMENTS 


COMPASS,  TRANSIT,  AND  LEVEL. 


BY 

A.  V.  LANE,  C.E.,  PH.D., 

ASSOCIATE  PROFESSOR  OF  MATHEMATICS, 
UNIVERSITY  OF  TEXAS. 


BOSTON: 

PUBLISHED  BY  GINN  &  COMPANY. 

1886. 


Entered,  according  to  Act  of  Congress,  in  the  year  1886,  by 

A.  V.  LANE, 
in  the  Office  of  the  Librarian  of  Congress,  at  Washington. 


J.  S.  CUSHING  &  Co.,  PRINTERS,  BOSTON. 


PREFACE. 


A  N  examination  of  those  text-books  in  which  this 
-*--*-  subject  is  or  should  be  treated,  reveals  the  fact 
that  it  is  for  the  most  part  very  meagrely  and  arbitrarily 
presented,  sometimes  dismissed  with  the  statement  that 
the  adjustments  should  be  made  by  the  maker  of  the 
instrument.  The  method,  when  given,  is  generally  with- 
out explanation  or  proof  that  it  will  accomplish  the 
desired  object,  so  that  the  student  must  either  take  the 
author's  word  for  it,  get  some  one  to  explain  it  to  him, 
or  work  it  out  for  himself ;  and  being  usually  unprepared 
for  such  original  work,  he  is  in  danger  of  adopting  the 
first  course  mentioned,  or  of  leaving  the  matter  in  doubt 
and  mystery. 

A  great  source  of  trouble  lies  in  the  fact  that  such 
authors  are  expected  to  express  themselves  in  accurate 
terms  and  do  not ;  the  word  "  half,"  for  example,  being  so 
often  used  for  that  which  is  at  best  but  approximately 
so,  that  the  student  marvels  at  the  talisman  for  such 
diverse  operations  being  so  uniformly  that  particular 
fraction. 

Perhaps  the  absence  of  explanation  and  proofs  of  the 


iv  PREFACE. 

correctness  of  some  of  the  methods  is  largely  due  to  the 
difficulty  of  making  the  subject  clear  to  those  who  have 
not  studied  Descriptive  Geometry.  It  is  believed,  how- 
ever, that  one  whose  attainments  in  the  line  of  mathe- 
matics go  no  further  than  through  Elementary  Trigo- 
nometry will  experience  no  difficulty  with  the  following 
discussion  of  the  adjustments  of  the  three  principal  in- 
struments used  by  surveyors  and  engineers. 

This  little  volume  has  been  called  forth  by  the  neea 
of  such  an  exposition  of  the  subject,  felt  by  the  author 
for  some  time  past  in  presenting  the  matter  to  his  classes 
in  Engineering,  and  any  suggestions  in  the  line  of 
improvement  will  be  acceptable. 

A.  V.  LANE. 
UNIVERSITY  OF  TEXAS, 
AUSTIN,  May,  1886. 


• 
i 


ABBREVIATIONS. 


A,  horizontal  axis  of  telescope  (Transit). 

B,  axis  of  the  level  bar  (Level). 

C,  line  of  collimation  (Transit,  Level). 

H  and  F,  horizontal  and  vertical  plane  of  reference. 

/,  intersection  of  A  and  C  (Transit). 

R,  plane  of  revolution  of  adjusted  C  (Transit). 

S,  axis  of  spindle  (Compass,  Transit,  Level). 

W,  intersection  of  cross-whcs  (Transit,  Level). 

Y  and  y,  longer  and  shorter  distances  from  B  to  the  centers 

of  the  telescope's  wye  bearings  (Level). 
U.,       level  tube  (Compass,  Transit,  Level). 
l.t.c.,     level  tube-case  (Compass,  Transit,  Level). 
Lt.cM.,  level  tube-case  axis  (Compass,  Transit,  Level). 


I.   THE  COMPASS. 


1.  The  Plate-Levels,  —  to  so  adjust  them  that  when  their 
'bubbles  are  centered,  the  plate  shall  be  horizontal. 


FIG.  1. 

Let  the  l.t.c.  GE  be  turned  in  a  vertical  plane  and 
about  its  center  G  through  an  angle.  The  bubble,  which 
was  at  Z>,  the  center  of  the  l.t.,  will  move  to  a  point  F' 
found  by  raising  a  vertical  through  the  new  position  of 
the  center  of  curvature  of  the  l.t.  The  arc  D'F'  through 
which  the  bubble  has  moved  subtends  the  angle 

02  =  DGD'  =  the  angle  between  DE  and  D'E' ; 


2  THE  COMPASS. 

i.e.,  the  motion  of  the  bubble  is  proportional  to  the  angle 
through  which  the  l.t.c.  is  turned  about  its  center. 

Let  us  suppose  that  the  l.t.c.a.  DiE1  is  not  parallel  to 
the  plate  JKi,  but  makes  with  it  an  angle  a.  Then,  when 
the  bubble  is  brought  to  the  center,  A-^i  being  horizontal, 
will  differ  a  from  horizontally.  If  now  the  plate 


D 


FIG.  2. 

be  turned  180°  on  $,  the  l.t.c.a.  will  have  changed  ends, 
and  will  lie  in  a  vertical  plane  parallel  to  the  one  which 
before  contained  it,  and  distant  therefrom  by  twice  the 
distance  of  the  l.t.c.a.  from  S.  Since  the  two  halves 
of  the  l.t.c.  are  precisely  alike,  the  effect  of  this  operation 
on  the  motion  of  the  bubble  is  clearly  the  same  as  that 
of  a  rotation  in  a  vertical  plane  and  about  the  center  of 
the  l.t.c.  through  an  angle  2 a.  If  then  the  bubble  is 


THE  COMPASS.  3 

brought  half-way  back  to  the  center  by  the  screws  at 
the  ends  of  the  l.t.c.,  and  the  rest  of  the  way  by  changing 
the  position  of  $,  the  l.t.c.a.  will  take  the  adjusted  posi- 
tion FG  (parallel  to  the  plate),  and  S  will  be  vertical, 
the  plate  also  horizontal.  See  Methods  of  Adjustment, 
p.  33. 

2.  The  Sights,  —  to  make  them  vertical  when  the  plate 
is  horizontal. 

The  plate  being  horizontal,  if  a  vertical  line  is  sighted 
to  and  each  sight  made  to  range  with  it  from  any  point 
of  the  other,  they  will  be  vertical.  See  Methods. 

3.  The  Needle,  —  to  so  adjust  it  that  any  one  vertical 
plane  may  contain  its  end  points,  its  center  of  rotation, 
and  the  center  of  the  graduated  circle. 

If  the  ends  of  the  needle  do  not  in  every  position  cut 
opposite  degrees,  (point  to  divisions  which  differ  180°), 
this  adjustment  is  needed.  Failure  to  do  so  is  due  to 
the  point  of  the  pivot  not  being  in  a  perpendicular  to 
the  plane  of  the  graduated  circle  at  its  center,  or  to  the 
needle  being  bent,  or  to  both  causes. 

Suppose  that,  one  end  of  the  needle  pointing  to  a  divis- 
ion G^  the  other  fails  to  point  to  its  opposite,  Ji-  Let 
the  pivot  be  bent  until  the  failure  is  corrected,  when  the 
conditions  will  be  such  as  are  represented  in  the  diagram. 
Reversing  the  sights  exactly,  the  points  F^  D^  E^  E± 
take  the  positions  F2,  D2,  E2,  EJ.  Evidently  the  divis- 


4  THE  COMPASS. 

ion  N2  is  approximately  half-way  between  3/2  and  G^ 
the  angle  F2D.,0  differing  slightly  from  J^DjO,  owing  to 
the  displacement  of  F±  and  F.2  from  0.  Hence,  if  the 
needle  be  so  bent  that  its  end  point  E.2  cuts  half-way  back 
to  G2J  its  ends  will  be  approximately  in  line  with  its 
center.  If  the  pivot  be  now  bent  until  the  needle  cuts 
the  readings  Jz  and  G.z,  F2  will  be  at  or  very  near  some 


FIG.  3. 


point  as  F3  in  the  line  GyJ*.  So  by  repeating  this  process 
the  ends  of  the  needle  will  be  brought  into  line  with  its 
center,  which  center  will  be  somewhere  in  the  line  G^J^  as 
at  F3. 

"When  this  has  been  done,  if  one  end  of  the  needle 
be  held  at  K2  (90°  from  J2)  and  the  pivot  bent  towards 


THE  COMPASS. 


K2L2  until  the-  other  end  cuts  L$,  F3  will  be  brought  into 
the  line  K2L2  and  nearer  to  its  proper  position. 

By  repeating  this  last  operation  on  the  pivot  with  refer- 
ence to  the  lines  G2J2  and  K2L^  alternately,  its  point 
will  be  adjusted.  • 


II.    THE   TRANSIT. 


1.  The  Levels,  —  Same  as  I.  1. 

2.  The  Needle.  —  Same  as  I.  3. 

3.  The  Line  of  Collimation,  —  to  make  it  perpendicular 
to  the  horizontal  axis  of  the  telescope,  so  that,  by  revolu- 
tion about  that  axis,  it  will  generate  a  plane  and  not  a 
conical  surface. 

It  should  be  remembered  that  C  is  the  line  of  sight 
and  is  determined  by  the  optical  center  of  the  object- 
glass  and  the  intersection  of  the  cross- wires.  The  wires 
should  be  respectively  vertical  and  horizontal ;  they  may 
be  made  so  by  causing  one  to  range  with  a  plumb-line 
or  known  vertical  line,  since  they  are  perpendicular  to 
each  other. 

Assume  a  horizontal  circle  with  center  /,  and  let  Ci 
differ  a  from  IE,  a  perpendicular  to  A.  Revolving  Ci 
about  A^  till  it  again  cuts  the  circle,  it  takes  the  position 
C2,  differing  a  from  ID,  and  therefore  180°  —  2  a  from  Ci. 
Turning  the  instrument  about  S  180°  — 2  a,  C2  becomes 
C3  in  coincidence  with  Ci,  while  A2  takes  the  position  A3. 
Revolving  C3  about  A3  until  it  again  cuts  the  circle,  it 
takes  the  position  (74,  differing  a  from  C5  perpendicular 


THE  TRANSIT.  7 

to  A5.  C5  and  ID,  respectively  perpendicular  to  A5  and 
AS,  differ  by  the  same  angle  as  those  positions  of  the 
axis;  viz.,  2a.  Since  the  arc  4  5  subtends  a,  5D  sub- 
tends 2  a,  and  D2  subtends  a,  the  ratio  of  the  arc  4  5  to 
the  arc  4  2  is  clearly  that  of  a  to  4  a,  or  one-fourth.  If  then 
C4  were  at  (75,  one-fourth  of  the  way  toward  (72,  it  would 


A.AI 
A\   ' 

2^ 

A              A, 

/? 

F\"   -A- 

\                         /^2? 

\      4  / 

\                 /l/4    / 

\ 

\  /  A 

4"x-  / 

\Ci 

\                        ,,'f 

^--*. 

\     ,,'''' 

^    i 

>        2 

FIG 

.  4. 

be  perpendicular  to  the  horizontal  axis  of  the  telescope, 
and  hence  in  adjustment.  In  practice  the  above  condi- 
tions cannot  usually  be  perfectly  realized.  Thus  the 
points  1,  2,  4,  5  are  not  in  one  plane,  which  introduces 
a  slight  error,  from  the  fact  that,  until  adjusted,  C revolves 
about  A  in  a  conical  surface  and  so  moves  to  the  right 
or  left  as  it  leaves  a  horizontal  position  ;  and  again,  the 


8  THE  TRANSIT. 

points  2,  4,  5  are  not  taken  exactly  on  a  circle,  but 
in  a  straight  line  on  top  of  a  stake.  The  errors  in  each 
case  are  small,  as  the  displacements  of  the  points  from  a 
horizontal  plane  and  the  distances  apart  of  the  last  three 
are  taken  quite  small  in  comparison  with  their  distance 
from  J. 

After  this  has  been  done,  it  may  be  found  that  the 
intersection  of  the  cross-wires  is  not  in  the  center  of  the 
field  of  view  of  the  eye-piece.  Since  the  position  of  O 
depends  only  on  the  object-glass  and  TF,  we  may,  with- 
out affecting  it,  move  the  e}Te-piece  by  the  proper  screws 
until  the  center  of  the  field  of  view  is  brought  into  coinci- 
dence with  the  intersection  of  the  cross- wires.  For  the 
eye-piece  is  simply  a  microscope  with  which  we  magnify 
and  invert  the  inverted  image  formed  at  the  cross-wires 
where  its  focus  is. 

Again,  it  is  evident  that,  even  when  C  is  not  perpen- 
dicular to  A,  we  may  locate  three  points  on  the  same  side 
of  the  instrument,  which  will  either  be  in  one  vertical 
plane  (in  line),  if  they  are  all  at  the  same  height  as  /; 
or  very  nearly  in  line,  if  they  do  not  vary  much  from 
that  height.  Let  these  points  be  1,  /  and  F  in  the  pre- 
ceding diagram.  Setting  up  the  instrument  at  the  middle 
one  7,  sighting  to  1 ,  and  turning  Ci  about  A±,  it  comes  to 
(72.  If  C2  were  perpendicular  to  J.2,  the  point  D  would 
be  sighted  to.  So  the  point  2  is  twice  as  far  from  F  as 
the  point  D  upon  which  to  effect  adjustment. 


THE  TRANSIT. 


9 


•  4.  The  Standards,  —  to  make  the  bearings  of  the  hori- 
zontal axis  of  the  telescope  equally  distant  from  the  plate, 
so  that  when  the  instrument  is  levelled,  the  line  of  collima- 
tion  will,  by  revolution  about  the  horizontal  axis  of  the 
telescope,  generate  a  vertical  plane. 


FIG.  5. 


Suppose  the  instrument  set  up  on  H,  level,  and  with  the 
preceding  adjustments  made.  Suppose  A  horizontal  and 
occupying  a  position  ID  parallel  to  H,  but  not  necessarily 
so  to  V.  Draw  the  vertical  IJ  and,  through  J,  the  line 


10  THE  TEANSIT. 

-ffilTa  in  H  parallel  to  ID.  Through  J  draw  JG  perpen- 
dicular to  K±K2 ;  and,  through  G,  GP8  perpendicular  to 
the  intersection  of  H  and  V.  Now  .R,  being  perpendic- 
ular to  A  or  ID,  is  perpendicular  to  H,  and  must  cut  V 
in  a  line  perpendicular  to  the  intersection  of  H  and  F. 
Hence  R  must  contain  IJ,  JG,  and  GPa.  Take  L  in 
6rP3  so  that  GL  =  JCr.  (7  may  be  directed  to  L,  when 
it  will  be  parallel  to  H  and  perpendicular  to  the  plane 
DIJ.  Since  the  arm  DI  of  the  right  angle  DIL  may  be 
rotated  about  IL  into  the  position  A±,  A  may  take  the 
position  AI  in  the  plane  DIJ,  and  C  be  still  fixed  upon  L. 
Since,  if  turned  into  that  plane,  C  will  take  the  position 
Zffi,  perpendicular  to  ^4X,  1?,  containing  K±  and  IL,  must 
now  cut  Hin  a  line  If^  parallel  to  IL  and  therefore  to 
JG,  and  so  must  cut  V  in  I^L. 

If  now  the  instrument  be  turned  on  S  or  IJ  through 
180°,  A±  takes  a  position  A2,  making  the  same  angle  with 
Z7as  before.  A2  being  in  the  plane  DIJ,  C  may  still  be 
directed  to  L  or,  in  that  plane,  to  IT2,  JK2  being  equal  to 
JK^  and  the  triangle  Jl/fj  having  turned  into  the  position 
JIK2.  R  must  therefore  now  cut  IT  in  a  line  K2F,  par- 
allel to  JG,  and  F  in  a  line  FL. 

Draw  PiP2  in  V  and  parallel  to  the  intersection  of  II 
and  F.  With  the  axis  in  the  position  AI,  C  could  be 
directed  to  Pt ;  with  the  axis  in  the  position  A2,  to  P2 ; 
and  in  the  adjusted  position  DI,  to  P3.  The  wires  should 
therefore  be  moved  toward  Pl  by  an  amount  P2P3. 

Since  JG  bisects  K^K^  it  bisects  EF,  and  therefore 


THE  TRANSIT. 


11 


GL  bisects  P2Pi-  So  the  wires  should  be  moved  half- 
way from  their  second  position,  P2,  toward  their  first,  P1? 
by  changing  the  height  of  one  of  the  standards. 

Practically,  the  ground   and  a  vertical  wall  take  the 
place  of  H  and  F.     (Given  last,  under  Methods,  II.  4.J 


FIG.  6. 

Suppose  the  instrument  at  7J,  the  right-hand  end  of  A 
the  higher,  and  R  occupying  the  position  P^P^K^.  Imag- 
ine the  instrument  turned  180°  on  $,  so  that  the  triangle 
occupies  the  position  IJK^  and  R  the  position 


12  THE  TEANSIT. 


,  just  as  in  the  preceding  case.  Directing  C  to  L19 
imagine  it  turned  to  the  left,  remaining  horizontal,  until 
a  point  L2  is  reached,  such  that,  on  revolving  C  about  A, 
it  will  again  cut  Px.  In  order  for  L2  to  be  such  a  point, 
it  is  evident  that  the  triangle  IJK2  must  have  revolved 
into  a  position  /J/ig,  such  that  perpendiculars  JP5  and 
/£jP4  to  JK3  shall  meet  the  intersection  of  H  and  V  in 
points  P5  and  P4,  respectively  on  a  vertical  through  L2 
and  on  the  prolongation  of  P3L2. 

C  was  first  turned  down  from  Pl  to  P2,  then  from 
P3  to  P4,  A  lying  in  the  plane  IJK3.  If  A  became 
horizontal  in  that  plane,  H  would  occupy  the  position 
L2P5J.  So  the  wires  should  be  moved  from  P4  to  P5, 
by  changing  the  height  of  one  of  the  standards.  We 
desire  therefore  to  determine  the  ratio  of  P4P5  to  P4P2. 

Refer  the  point  J  to  P2  by  JG  =  d,  angle  JGP2  =  0, 
GP2  =  b.  Let  L2P5  =  LlG  =  IJ=i.  Let  P,0=p, 
and  the  angle  I£3P4P5  =  angle  JP5G  —  <£.  Then  the  an- 
gle G  JP5  =  6  —  <£  ;  also 

P5N=  JK3  =  JK2  =  JKi  =  MP2  =  b  sin  0. 
By  similar  triangles 

/IN 


P40     p     P20 

By  alternation 


P2G      P20' 

and  this,  by  composition,  gives 


THE   TRANSIT.  13 

/ 


or 


and  in  the  triangle  P5GJ 


d  sin  <£  tan 

whence,  by  (3), 


6 
=  P4P2  -  (P4P5  +  &)  = 


Whence 

PJP,  =  Pft-L-b.  (2) 

In  the  triangle  NP±P5  we  have 


and 

(2). 


(4) 


Substituting  these  in  (4)  and  clearing  of  fractions,  we 
have 

b  •  P±P2  (p  -  i)  4-  bdp  cos  6 


i2  -  2  Up  •  P4P2  +  &y  -  6V  sin20. 


14  THE   TRANSIT. 

Squaring,  cancelling,  and  taking  out  the  common  factor 
P4P2,  we  find 


P  p  =  2  bdp  [di  +  b  (p  ^-  J)  cos  0] 
"  "~&?-V(p-i)* 

Substituting  this  in  (2),  we  obtain 


P4P2     p 
the  required  ratio. 

If  0  be  90°  and  p  —  i  =  d,   this  ratio  becomes 

i      i2-b2 
p        2ip 

b2  i 

or,  neglecting  the  small  fraction  —  ,  it  becomes  -- 

2  ip  2p 

These  conditions  are  usually  very  nearly  satisfied,  so 
we  may  consider  -  -  quite  an  approximate  value  of  the 
true  ratio.  Assuming  i  =  5  ft.  and  p  =  50  ft.  as  about 
average  values,  this  ratio  equals  —  ;  while,  even  if  b 

b2 

could  equal  6  inches,  the   neglected  fraction  -  —  -  would 

2ip 

only  equal  -   —  •     (Given  second,  under  Methods,  II.  4.) 

Suppose  that,  having  as  before  located  the  point  P2 
from  the  point  PI,  we  turn  the  instrument  about  half- 
way around  on  S  and,  revolving  the  telescope,  fix  the 
wires  again  on  the  lower  point  P2.  Now,  raising  them 


THE  TEANSIT. 


15 


to  P4  at  height  of  P19  we  wish  to  determine  the  ratio  of 
P4P6  to  P4PX. 

For,  as  before,  R  having  first  the  position  P^/ii,  on 
turning  exactly  half-wa}*  around,  it  would  take  the  posi- 
tion L^FK^ ;  and,  directing  C  to  £1?  we  must  turn  the 


FIG.  7. 

instrument  to  the  right  until  C  strikes  a  point  Z/2,  such 
that,  on  revolving  about  A,  the  point  P2  is  again  cut. 

The  triangle  IJjfa  has  now  revolved  into  the  position 
IJKz,  and  R  into  the  position  P4P3KB ;  while,  if  A  be- 
came horizontal,  R  would  have  the  position  P5G2J. 


16  THE  TRANSIT. 

The   references   of  J  to  P2   are   now  JGl  =  d,    angle 
J6?1P2  =  6»,    G1P2  =  b. 

The  angle  JG20  is  <£,  and  the  angle  G^G2  =  <j>  —  Q. 
Since,  by  similar  triangles, 


P  G 

and  we  will  first  determine  ~ 


OP3 

By  alternation, 


and  this,  by  composition,  gives 


or, 


__ 

4px    oq      i     OQ  ' 


(1) 


Whence 

P3G2=OQ.l--b.  (2) 

P 

In  the  triangle  NG2P3  we  have 
_ 


THE  TRANSIT.  17 


and,  in  the  triangle 


A  =  sin  (<£  -  Q)  =  cQ£         sin  0 


d  sin  <£  tan  </> 

whence,  by  (3), 


Ct 


(4) 


and 

PA_°«^Z*  by(2). 

Substituting  these  in  (4)  and  clearing  of  fractions,  we 
have 
bdpcosO  —  bi-  OQ 


i2-2  Up  -  OQ  +  b2p2  -  by  sin2  0. 

Squaring,  cancelling,  and  taking  out  the  common  fac- 
tor OQ,  we  find 

d  —  b  cos  0 


Substituting  this  in  (2)  ,  we  obtain 


OQ      p      2dp     d  —  bcosOp       2d(d  — 
Therefore 


i          OQ        2dp  d-bcosO 

the  required  ratio. 


18  THE  TRANSIT. 

If  0  be  90°,  neglecting  the  small  fraction 

(P  ~  0  & 

2d2p 
it  becomes 

p  —  i  _  1  _  i_ 
2p  ~2     p' 

or  approximately  -,  since  —  is  usually  about  —  ;  while 
the  neglected  fraction,  under  the  preceding  conditions,  is 

.     (Given  first,  under  Methods,  II.  4.) 

18000 

This  last  is  the  best  and  most  generally  used  method. 
The  first  is  excellent,  if  divisions  on  the  horizontal 
circle  such  as  the  zeros  may  be  depended  on  as  exactly 
opposite.  In  the  second,  the  amount  through  which  the 
wires  are  to  be  moved  for  correction  is  such  a  small  part 
of  a  small  distance  that  it  is  difficult  of  exact  application. 

5,  a.  The  Vertical  Circle's  Vernier  Zero,  —  to  so  ad- 
just it  that  when  the  vertical  circle's  zero  is  in  line  with  it, 
the  line  of  collimation  shall  be  horizontal. 

If,  instead  of  a  full  circle  firmly  attached  to  A,  there 
is  only  an  arc  of  a  circle,  which  may  be  turned  upon  and 
clamped  to  it,  as  is  the  case  in  some  instruments,  this 
adjustment  depends  upon  and  is  made  after  that  of  the 
level  attached  to  the  telescope.  The  method  of  making 
it  in  that  case  is  explained  further  on  (after  6) . 


THE   TRANSIT. 


19 


Let  us  suppose  that  the  vernier  zero  Z^  is  not  in  its 
proper  position  (vertically  below  the  center  of  the  verti- 
cal circle)  ;  so  that,  when  the  circle  zero  Zi  is  made  to 
coincide  with  it,  C  has  the  position  Ci,  making  the  same 
angle  with  the  horizontal  OiD1  that  Ofa  makes  with  the 
vertical  0^. 

Let  Pl  be  some  point  in  Ci.  Turning  the  instrument 
180°  on  S,  Ci,  PU  zx,  Zi  move  to  <72,  P2,  z2,  Z2.  The 


angle  P202D2  being  equal  to  y,  if  O2  be  turned  on  ^42 
through  180°  — 2 y,  it  takes  the  position  (73,  and  z2  the 
position  2;3,  the  angles  P&03D3  and  E20szs  being  each 
equal  to  y.  The  division  zs'  opposite  ^3  is  now  y  from  F2, 
and  2y  from  ^3.  Hence  if  Z2  were  moved  half-way  to  z3' 
and  then  «3'  brought  into  coincidence  with  it,  (73  would 


20  THE   TRANSIT. 

take  the  horizontal  position  037>3,  and,  the  zeros  being 
also  in  coincidence,  the  adjustment  would  be  effected. 
If  A3  were  turned  into  the  position  A4<  and  C3  directed  to 
P4,  the  angle  P±0±D±  being  slightly  different  from  y,  the 
ratio  corresponding  to  the  above  would  not  be  exactly 
one-half,  yet  by  taking  the  distance  of  P1  from  the  instru- 
ment sufficiently  great  in  comparison  with  A,  the  point  Px 
may  be  used  for  P3  with  but  little  error  ;  and  it  is  so  used, 
the  process  being  repeated  until  the  whole  error  becomes 
inappreciable. 

6.  a.  The  Level  attached  to  the  Telescope,  — to  so  ad- 
just it  that  when  its  bubble  is  at  the  center,  the  tine  of 
collimation  shall  be  horizontal. 

A  full  vertical  circle  being  present,  and  its  zero  in 
coincidence  with  the  vernier  zero  after  the  preceding 
adjustment,  if  the  bubble  is  brought  to  the  center  by 
means  of  the  nut  at  either  end  of  the  level,  the  adjust- 
ment is  effected. 

5.  b.  The  Level  attached  to  the  Telescope,  —  If,  how- 
ever, the  instrument  has  the  movable  arc,  the  level  is 
first  adjusted  as  follows  : 

Let  J9,  J],  -E7,  and  Js  be  four  points  in  line,  and  in  the 
order  named  ;  also  equally  distant  horizon  tall}'.  Suppose 
the  instrument  set  up  at  «/i,  and  Cl  directed  to  a  graduated 
rod  held  on  the  stake  Z>,  giving  a  reading  T^Z).  Turning 
the  instrument  180°  about  £,  without  otherwise  disturbing 


THE   Til  AX  SIT. 


21 


Ci,  C2  will  cut  the  rod  held  on  the  stake  E  at  a  reading 
F.2E,  the  points  F±,  F2  being  at  the  same  height,  since 
Cj^  and  C'2  make  equal  angles  with  the  vertical  I^J^  and 


the   positions  of    the   rod  are  equidistant  from  7j.     Let 
the  instrument  be  now  set  up  at  /3,  and,  with  C  in  a  posi- 


22  THE  TRANSIT. 

tion  <73,  the  readings  F3E  and  F4D  noted.     Draw 
parallel  to  F2F^  and  therefore  horizontal. 

F^  =  F4D-  FJ)  and  F&F2  =  FSE  -  F2E  being  known, 
let  x  =  LFl  =  KF2,  and  we  have,  by  similar  triangles, 

-  x     LI      3 


whence 

_ 

JU  — 

is  known. 

So,  by  directing  C  to  a  reading  Z^  4-  a?,  it  takes  the 
horizontal  position  (74,  and  the  level  on  the  telescope  may 
be  adjusted  by  bringing  its  bubble  to  the  center,  by  means 
of  the  nut  at  either  end. 

6.  b.  The  Vertical  Arc's  Vernier  Zero.  —  The  vertical 
arc  may  now  be  turned  on  A  and  clamped  with  its  zero  in 
coincidence  with  the  index  zero,  effecting  its  adjustment. 

For  convenience  of  application  as  to  sign,  etc.,  it  is  well 
to  take  the  first  reading  just  above  the  top  of  the  higher 
of  the  stakes  D,  E,  and  to  set  up  at  J3  so  that  J3  shall  be 
higher  than  F^  F2  ;  taking  the  readings  F&E,  F4D  greater 
than  F2E,  FJ)  respectively. 

By  driving  the  stakes  at  D  and  E  so  that  their  tops  are 
at  Fl  and  F^  the  first  two  readings  reduce  to  zero,  and 
F3F2,  F^  are  the  only  readings  taken  ;  they  are  used  as 
stated  above,  x  being  itself  the  reading  of  the  target  at 
FI  for  horizon  tali  t}T. 


III.    THE   LEVEL. 


1.  The  Line  of  Collimation,  —  to  so  adjust  it  that  it 
will  pass  through  the  centers  of  the  circles  of  the  wye  bear- 
ings of  the  telescope. 

This  adjustment  is  a  first  step  toward  making  C  per- 
pendicular to  S. 


FIG.  10. 

Let  D  and  E  be  the  centers  of  the  circles  of  the  wye 
bearings  and  Cl  the  position  of  C  sighting  to  some  point 
as  P!,  distant  PiP3  from  the  line  ED  produced.  If  now 
Cl  be  turned  180°  about  ED,  it  will  take  the  position  <72, 
making  with  ED  an  angle  equal  to  that  made  by  Ci,  and 
sighting  to  some  point  as  P2  at  the  same  distance  as  P1 
from  0  and  P3.  Clearly  now  for  C,  to  coincide  with  ED, 
W.2  must  be  brought  to  TFg,  half-way  back  to  TFi,  its  for- 
mer position,  by  moving  it  until  C2  takes  the  position  (73 
sighting  to  a  point  P3,  half-way  from  P2  towards  Plf 
This  may  be  best  accomplished  by  adjusting  each  wire 
in  succession,  if  they  be  much  in  error,  by  means  of  some 


24  THE   LEVEL. 

line  sighted  to,  until  each  is  nearly  right,  then  completing 
the  adjustment  of  each.  This  amounts  to  substituting 
for  P!  a  line  through  Px  and  perpendicular  to  the  plane 
of  the  paper,  for  P2  a  like  but  imaginary  line,  and  for 
WH  Wz,  and  TF^one  of  the  cross-wires,  also  perpendicular 
to  the  plane  of  the  paper  and  through  those  points. 

As  already  explained  in  II.  3,  the  center  of  the  field  of 
view  should  now  be  brought  into  coincidence  with  W  by 
moving  the  proper  screws,  and  the  correctness  of  the 
centering  may  be  tested  by  turning  the  telescope  in  the 
wyes,  when  the  object  should  not  appear  to  shift  its  posi- 
tion. 

2.  The  Level,  —  to  make  the  level  tube-case  axis  par- 
allel to  the  line  of  collimation. 

In  the  explanation  of  I.  1  it  was  shown  that  in  the 
operation  of  reversing  the  l.t.c.a.  with  reference  to  any 
plane  or  line  to  which  its  ends  were  referred,  the  bubble 
moves  from  the  center  in  the  first  position  to  some 
point  in  the  second  position,  such  that  the  arc  between 
the  center  and  that  point  is  double  the  arc  through  which 
the  bubble  should  be  moved  back  in  order  to  make  the 
l.t.c.a.  parallel  to  the  line  or  plane  of  reference.  And 
this,  too,  whether  that  line  or  plane  be  horizontal  or  not. 

Since  now  C  contains  the  centers  of  the  circles  of  the 
wye  bearings,  by  reversing  the  telescope  end  for  end  as 
to  the  wyes,  we  may  effect  this  adjustment ;  the  ends  of 
the  l.t.c.a.  being  referred  to  C.  But  there  is  a  disturbing 
element  to  be  considered  in  this  connection. 


THE   LEVEL. 


25 


The  l.t.c.  has  a  screw  at  one  end  for  lateral  adjustment, 
to  bring  the  Lt.c.a.  into  one  plane  with  (7,  and  another, 
at  the  other  end,  for  vertical  adjustment  to  make  these 
lines  equidistant  throughout.  If  we  remove  the  clips  from 
the  wyes,  and  reverse  the  telescope  end  for  end,  it  may 
be  that  we  do  not  put  it  down  with  the  vertical  screw 
immediately  below  (7,  as  before,  but  slightly  rotated  to 
one  side.  We  must  therefore  examine  the  effect  on  the 
bubble  of  a  slight  rotation  of  the  Lt.c.a.  about  0,  with  a 
view  to  ascertaining  how  much  if  any  of  the  bubble's  mo- 
tion on  reversal  is  due  to  this  cause. 


J) 


FIG.  11. 

Imagine  the  plane  of  the  paper  vertical,  and  containing 
C  and  the  Lt.c.a.  Draw  DE  tangent  to  the  l.t.  at  E. 
The  highest  point  of  the  l.t.  being  that  of  contact  with  a 
horizontal  tangent  plane,  it  is  evident  that,  0  being  hori- 
zontal, this  highest  point  will  lie  on  a  cross-section  of  the 
l.t.  at  E  for  any  revolution  less  than  90°  about  (7, 
whether  the  Lt.c.a.  is  equidistant  from  Cor  not.  For  its 
position  does  not  affect  the  symmetry  of  the  parts  of  the 
l.t.  on  each  side  of  E,  but  simply  their  amounts. 


26  THE   LEVEL. 

Suppose  C  is  not  horizontal.  After  a  revolution  of 
90°,  #iGy  and  KJQ  turning  about  G2'  and  Ay  respec- 
tively, into  a  horizontal  position,  G1  will  be  at  the  same 
height  as  6r2',  and  therefore  higher  than  K^  will  be  (same 
height  as  K.2') .  So  the  bubble  will,  as  the  rotation  begins, 
start  towards  the  end  G- 


FIG.  12. 

Again,  in  Fig.  11,  suppose  L  to  be  slightly  in  front  of 
the  plane  of  the  paper,  and  F  a  like  amount  behind.  If 
we  rotate  towards  us,  L  rises  and  F  falls,  the  bubble 
therefore  running  toward  /f,  instead  of  remaining  at  E. 
as  before ;  while,  if  we  rotate  the  other  way,  the  reverse 
occurs.  Also,  in  Fig.  12,  if  we  make  a  like  supposition, 
since  the  bubble  before  moved  toward  G^  whichever 
way  rotation  took  place,  while  the  effect  of  this  new  sup- 
position tends  to  make  it  move  towards  KI  or  GI  (accord- 
ing as  we  rotate  towards  us  or  away) ,  it  is  clear  that  these 
causes  may  combine  to  move  the,  bubble  one  way,  or  may 
oppose  and  even  neutralize  each  other. 

Thus  it  is  evident  that,  knowing  nothing  of  the  actual 
state  of  the  positions  of  C  and  the  l.t.c.a.,  we  cannot 


THE  LEVEL. 


27 


predicate  anything  as  to  how  much  of  the  motion  of  the 
bubble  on  reversal  may  be  due  to  the  cause  just  exam- 
ined. Nevertheless  the  lateral  screw  may  be,  by  trial-,  so 
adjusted  as  to  eliminate  the  effect  of  this  slight  rotation, 
and  finally  to  bring  the  Lt.c.a.  into  the  same  plane  with 
C,  as  in  the  method  given  farther  on. 

3.  The  Wyes,  —  to  so  adjust  them,  as  to  the  distances 
of  the  centers  oj  their  circular  bearings  from  the  axis  of 
the  level-bar,  that  the  axis  of  the  spindle  shall  be  perpen- 
dicular to  the  line  of  collimation. 

B  is  supposed  to  be  made  perpendicular  to  S,  so  that 
if  Y  be  made  equal  to  y,  C  will  be  parallel  to  J5,  and 
therefore  perpendicular  to  S.  But  if  B  differs  by  some 
angle  /3  from  perpendicularity  to  S,  by  reason  of  bad 
construction  or  some  strain  received,  Y  and  y  must  be 
given  such  values  as  to  counteract  this  and  make  C  per- 
pendicular to  S. 


FIG.  13. 


28 


THE  LEVEL. 


From  the  diagram  it  is  easily  seen  that  the  necessary 
condition  is  a  =  /5,  with  opposing  effects  on  the  bubble 
(a  special  case  of  which  is  a  =  0,  /2  =  0,  when  G  is 
parallel  to  B  as  well  as  perpendicular  to  S) ,  or,  in  other 
words,  C  making  the  same  angle  with  B  that  B  does  with 
a  perpendicular  to  $,  and  S  making  an  acute  angle  with 
that  half  of  B  which  carries  Y. 


2/3 


FIG.  14. 


Let  us  suppose  S  to  have  a  position  Si,  making  an 
acute  angle  90°  —  p  with  the  y  half  of  B.  We  see  that, 
by  reversing  about  S^  B±  and  Ci  take  the  positions  B3 
and  Ci;  while  if  St  had  the  position  (S2)  perpendicular 
to  BI,  B±  and  Ci  would  take  the  positions  (B.2)  and  (O2), 
on  reversal.  Rotating  (Ba)  through  2/3  into  the  position 
BS,  (C2)  rotates  through  an  equal  angle  into  the  position 


THE   LEVEL.  29 

v  / 

C3.     Since,  then,  C3  makes  an  angle  2/5  with  (O2),  and 
(<72)  makes  2  a  with  d,  C3  makes    2  a  +  2/5   with  d. 

This  may  also  be  shown  thus :  Drawing  DE  parallel  to  O3 
and  DF  parallel  to  J?3,  we  have  EDF=  a.  But  FDG  =  20,  since 
J^D  is  parallel  to  B3  and  D<7  to  Bx.  Therefore  the  (acute)  angle 
between  D F  and  ( (72)  equals  2  #  —  o.  Hence  the  (acute)  angle 
between  DE  and  (<72)  or  C3  and  (O,)  equals  a +  2)8  —  a  =  20; 
whence  the  (acute)  angle  between  C3  and  <7X  equals  2  a+  2/3. 

Thus  we  see  that,  on  reversing  about  S^  the  bubble 
moves  from  the  center  toward  Y  over  an  arc  corresponding 
to  2  a +  2/5,  the  effects  of  a  and  /5  being  in  this  case 
cumulative.  Remembering  that  only  2  a  of  this  is  due 
to  (7s  not  being  parallel  to  -B,  and  that,  if  the  bubble 
were  moved  back  over  a,  C3  would  be  parallel  to  B3,  we 
see"  that  if  we  shorten  Y  until  the  bubble  comes  half-way 
back  to  the  center,  we  move  it  over  i(2a-}-2/5)  =  a-r-/5; 
i.e.,  /5  more  than  the  proper  amount  (a)  to  make  O3  parallel 
to  B3.  Thus  we  have  replaced  the  error  a  by  another,  /5, 
and  made  that  wye  which  was  the  longer,  now  the 
shorter ;  having  decreased  Y  too  much  for  parallelism 
of  C3  and  J53.  Since  Y  and  y  have  interchanged  ends, 
8  now  makes  an  acute  angle  with  the  Fhalf  of  jB,  and 
the  angle  between  O3  and  _B3  is  equal  to  that  between  B 
and  a  perpendicular  to  $,  —  precisely  the  conditions  of 
Fig.  13  (except  that  S  is  not  vertical).  So  that  the 
adjustment  is  effected  with  B  not  perpendicular  to  S. 

Suppose  the  effects  of  a  and  /5  on  the  bubble  are  oppos- 


30 


THE   LEVEL. 


ing,  that  a  is  greater  then  (3  (S  making  an  acute  angle 
with  the  ]Thal^  of  J5).  Reversal  about  Si  brings  Z>\  and 
Ci  into  the  positions  B3  and  C3 ;  while  revel-sal  about  (S2) 
perpendicular  to  BI  would  have  brought  them  into  the 
positions  (B2)  and  (C2).  Introducing  the  effect  of  /?, 


FIG.  15. 

(<72),  which  had  separated  from  Ci,  returns  toward  but 
not  to  it,  since  a  >  /?.  The  combined  effect  of  a  and  /? 
is  thus  to  move  the  bubble  over  an  arc  corresponding  to 
2a  —  2/3  toward  T^.  Since,  in  bringing  it  half-way  back 
to  the  center,  it  is  moved  over  a  —  (3  (ft  less  than  the 
proper  amount  to  make  (73  parallel  to  J33) ,  there  remains 
an  error  ft  between  C  and  B.  But  since  Y3  has  not  been 
made  shorter  than  y8,  S  is  still  making  an  acute  angle 
with  the  Fhalf  of  B,  and  so  the  adjustment  is  effected, 
as  in  Fig.  13. 


THE  LEVEL.  31 

Finally,  suppose  a  and  /5  opposing,  and  /2>a.  The 
effect  of  a  being  to  make  the  bubble  run  toward  the  (  Y2) 
end  of  (C4),  and  fi  causing  (C2)  to  turn  back  through 
Oi  to  C3  (making  the  bubble  run  away  from  the  Y3  end 
of  (73),  their  combined  effect  will  move  it  2/3  —  2  a  toward 
the  Ys  end  of  O3.  To  move  it  half-way  back  we  must 
now  lengthen  Y^  (or  shorten  ?/3),  and  so  increase  a  by 
-|-(2/3  —  2a)  ;  i.e.,  by  /?  —  a,  making  the  angle  between 


-B3  and  03  a  +  (/8  —  a)  =  /3,  >S  still  making  an  acute 
angle  with  the  Y  half  of  J5,  —  again  the  conditions  of 
Fig.  13. 

Thus  we  see  that,  when  a  and  ft  are  cumulative  in 
effect,  the  process  results  in  making  them  opposing,  and 
if  they  are  at  first  opposing,  they  result  opposing.  For, 
in  the  first  case,  we  make  the  longer  wye  the  shorter,  thus 
introducing  a  change  ;  while  in  the  latter,  although  the 
length  of  one  of  the  wyes  is  changed,  the  longer  one 


32  THE  LEVEL. 

remains  the  longer.  In  each  case  a  is  changed  to  the 
amount  /3  (there  being  no  facilities  for  changing  /3),  aii'd 
so  they  are  left  equal  and  opposing. 

It  should  be  noted  that,  while  C  is  made  perpendicular 
to  £,  by  thus  bringing  the  bubble  half-way  back  to  the 
center,  it  must  be  brought  the  rest  of  the  way  back  by 
the  leveling-screws ;  then  C  will  be  horizontal,  and  if  it  is 
made  so  in  two  intersecting  positions,  S  will  be  vertical. 


METHODS   OF  ADJUSTMENT. 

I.  THE  COMPASS. 

1.  The  Levels.  —  Set  up  the  instrument,  and  bring  the 
bubbles  'to  the  center  by  pressure  of  the  hands   on  the 
plate.     Reverse  the  sights,  and  if  the  bubbles  remain  at 
the  center,  the  levels  are  in  adjustment.     If  the}"  do  not, 
bring  each  half-way  back  to  the  center  by  means  of  the 
screws  at  the  ends   of   the   level   tube-case,  the  rest  of 
the  way  by  means  of  the  plate,  and  repeat.     Or  perform 
this  operation  with  one  level  at  a  time  until  it  is  nearly 
adjusted,  then  with  the  other,  finally  completing  the  adjust- 
ment of  each  and  seeing  that  both  will  reverse  correctly 
and  remain  in  the  center  during  an  entire  revolution  of 
the  plate. 

2.  The   Sights. — Observe   through  the   slits   a   good 
plumb-line   and   if   either   sight   fails  to   range  with   it, 
make  it  do  so  by  whatever   means  the  instrument  calls 
for,  —  usually,  filing  a  little  off  of  one  side  of  the  surface 
of  contact  of  the  sight  with  the  plate. 

3.  The   Needle. — If  the   needle   will   not  in   various 
positions  cut  opposite  degrees,  this  adjustment  is  needed. 


34  METHODS   OF  ADJUSTMENT. 

Having  removed  the  glass  top  of  the  compass-box,  with 
a  splinter  of  wood  bring  one  end  of  the  needle  to  any 
prominent  graduation,  as  the  zero,  having  the  eye  nearly 
in  the  plane  of  the  graduated  circle,  and  see  if  the  other 
end  corresponds  to  the  opposite  division.  If  not,  bend 
the  center-pin  (pivot)  with  a  small  wrench,  about  one- 
eight  of  an  inch  below  its  point,  until  that  other  end  of 
the  needle  will  cut  the  other  zero.  Reverse  the  zeros 
but  not  the  ends  of  the  needle,  and,  holding  with  the 
splinter  the  same  end  of  the  needle  at  the  new  zero, 
note  what  division  the  other  end  cuts.  Bend  the  needle 
until  that  end  cuts  a  division  half-way  back  to  the  adja- 
cent zero.  This  puts  the  needle's  ends  very  approxi- 
mately in  line  with  its  center.  Bend  the  center-pin  again, 
until  the  ends  of  the  needle  will  cut  the  zeros.  Repeat 
until  perfect  reversion  is  obtained. 

Bring  one  end  of  the  needle  to  the  90°  division,  and 
if  the  other  end  does  not  cut  the  opposite  division,  bend 
only  the  pivot  until  it  does,  and  repeat,  using  alternately 
the  line  of  zeros  and  that  of  the  90°  divisions,  until  it 
will  cut  opposite  degrees  in  any  position. 

II.   THE  TRANSIT. 

1,  The  Levels.  —  Adjust  by  reversal  as  in  I.  1,  accom- 
plishing the  reversal  by  means  of  the  readings  of  the 
horizontal  circle  and  moving  the  plate  by  means  of  the 
leveling-screws. 


METHODS   OF   ADJUSTMENT.  35 

2.  The  Needle.  —  Same  as  I.  3. 

3.  The  Line  of  Collimation.  —  Having  the  instrument 
set  tip  and  leveled  on  tolerably  level  ground,  make  the 
wires   respectively  horizontal   and  vertical   by  loosening 
the  proper  screws  and  turning  the  ring  around  until  the 
vertical  wire  ma}7  be  made  to  coincide  with  some  known 
vertical   line  as  a  plumb-line,  or  the  vertical  edge  of  a 
building  from  two  to  five  hundred    feet   distant.     Make 
the  screws  tight  again. 

Select  or  locate  a  point  from  two  to  five  hundred  feet 
distant,  clamp  the  plates,  revolve  the  telescope  on  its 
axis,  and  locate  a  point  on  a  stake  on  the  other  side  of 
and  at  the  same  distance  from  the  instrument  as  the 
first.  Uiiclamp  the  plates  and  turn  about  the  spindle 
until  the  wires  can  be  again  fixed  on  the  first  point. 
Clamp  the  plates  and  again  revolve  the  telescope  on  its 
axis.  If  the  intersection  of  the  wires  strikes  the  second 
point,  the  line  of  collimation  is  in.  adjustment.  If  not, 
the.  intersection  of  the  cross-wires  should  be  brought 
one-fourth  of  the  way  back  to  this  second  point  by 
means  of  the  pair  of  cross-wire  screws,  on  the  sides  of 
the  telescope,  which  move  the  vertical  wire. 

The  operator,  in  loosening  one  of  these  screws  and 
tightening  the  other,  should  remember  that  they  have 
their  bearings  in  the  cross-wire  ring,  that  a  non-inverting 
telescope  inverts  the  relations  of  the  cross-wires  to  the 
object,  and  vice  versa.  He  must  therefore,  in  a  non- 
inverting  telescope,  proceed  as  if  apparently  to  move 


36  METHODS   OF  ADJUSTMENT. 

the  vertical  wire  in  the  opposite  direction  from  that 
desired. 

Test  by  repetition. 

If,  when  this  has  been  done,  the  intersection  of  the 
cross-wires  is  not  in  the  center  of  the  field  of  view,  move 
the  latter  until  they  are,  by  means  of  the  screws  which 
control  the  eye-piece,  loosening  and  tightening  them  in 
pairs  ;  the  movement  being  now  direct  or  as  it  appears 
it  should  be. 

Another  method  is  to  locate  three  points  in  line  and 
all  on  the  same  side  of  the  instrument.  Then  setting  up 
over  the  middle  point,  sight  to  one  of  the  end  ones,  and 
clamping  the  plates,  revolve  the  telescope  to  sight  to 
the  other  end  one.  If  the  intersection  of  the  wires  fails 
to  strike  it,  move  that  intersection  half-way  to  the  point 
by  means  of  the  vertical  wire,  as  just  explained,  and 
repeat.  Then  center  the  eye-piece  as  in  the  preceding 
method. 

4.  The  Standards.  —  Select  a  tolerably  level  piece  of 
ground  in  front  of  a  tall  spire,  tower,  or  like  object,  that 
shall  afford  from  top  to  base  a  long  range  in  a  vertical 
direction.  Set  up  and  level  the  instrument,  so  that  it 
will  be  about  as  far  in  front  of  the  structure  as  its 
telescope  is  below  a  good  sight-point  near  the  top. 
Clamp  to  the  spindle,  and,  fixing  the  wires  on  the  point 
selected,  clamp  the  plates  and  lower  the  wires  to  some 
point  found  or  marked  at  the  base  of  the  structure.  (If 


METHODS   OF  ADJUSTMENT.  37 

the  ground  is  not  very  level,  take  the  point  in  the  face  of 
the  building,  and  at  about  the  height  of  the  ground  on 
which  the  instrument  stands.)  Unclamp  the  plates,  and, 
turning  the  instrument  about  half-way  around,  revolve 
the  telescope,  and  again  fix  the  wires  on  the  lower 
point.  Clamp  the  plates  and  raise  the  wires  to  the 
height  of  the  upper  point.  If  they  cut  it,  the  standards 
are  in  adjustment.  If  they  do  not,  bring  them  half- 
way to  it,  by  raising  the  right-hand  end  (or  lowering 
the  left)  of  the  horizontal  axis  of  the  telescope,  if  the 
wires  are  to  the  right  of  the  point ;  by  raising  the  left 
(or  lowering  the  right),  if  they  are  to  the  left.  Most 
instruments  have  a  means  of  making  this  adjustment 
at  one  end  of  the  horizontal  axis,  —  a  movable  bearing. 
If  the  instrument  has  no  such  means,  file  equally  a  little 
off  of  the  feet  of  the  higher  standard.  Repeat  until  the 
adjustment  is  perfected. 

Another  method  is  to  sight  to  an  upper  point,  and 
lowering  the  wires,  fix  a  lower  point  at  the  base  of  the 
structure  just  as  before  ;  but,  on  turning  the  instrument 
about  half-way  around  and  revolving  the  telescope,  fix 
the  wires  again  on  the  upper  point.  Clamping,  and 
lowering  the  wires  to  the  height  of  the  lower  point,  if 
they  do  not  strike  it,  move  them  back  towards  it  over 
that  fractional  part  of  the  distance,  which  the  height  of 
the  instrument  above  the  ground  is  of  double  the  height 
of  the  upper  point  above  the  ground.  This  is  to  be 
done  by  raising  the  right-hand  end  (or  lowering  the  left) 


38  METHODS   OF   ADJUSTMENT. 

of  the  horizontal  axis,  if  the  wires  strike  to  the  left  of 
the  point ;  raising  the  left  (or  lowering  the  right) ,  if  they 
come  to  the  right.  Repeat  until  the  adjustment  is 
perfected. 

Another  method  (which  is  dependent  on  the  accuracy 
of  the  graduation  of  the  horizontal  circle)  is  as  follows : 
Setting  up  and  leveling  the  instrument,  clamp  the  plates 
at  zero  or  some  other  convenient  reading,  and  fix  the 
wires  on  an  elevated  (or  depressed)  point  of  sight, 
clamping  to  the  spindle.  Unclamp  the  plates,  and, 
turning  through  exactly  180°,  as  shown  by  the  horizontal 
circle  reading,  clamp  again.  If  on  now  raising  (or 
lowering)  the  wires  to  the  point,  they  cut  it,  the  stand- 
ards are  in  adjustment.  If  they  do  not,  bring  them  half- 
way to  it,  by  changing  the  height  of  one  of  the  standards 
in  such  a  way  as  to  raise  the  right-hand  end  (or  lower 
the  left)  of  the  horizontal  axis,  if  the  wires  strike  to  the 
right  of  the  point ;  raising  the  left  (or  lowering  the  right) , 
if  they  come  to  the  left. 

5.  a.  The  Vertical  Circle.  —  (If  only  an  arc  is  present, 
see  56.)  Bring  its  zero  into  coincidence  with  the  zero 
of  the  vernier  attached  to  the  standards,  and  with  the  tele- 
scope find  or  place  some  point  or-  horizontal  line  cut  by 
the  horizontal  wire  and  about  two  or  three  hundred  feet 
distant.  Turn  the  instrument  about  half-way  around, 
revolve  the  telescope,  and  fixing  the  wires  upon  the 
point,  or  the  horizontal  wire  upon  the  line  first  selected, 


METHODS   OF   ADJUSTMENT.  39 

clamp  the  telescope  and  note  if  the  zeros  are  again  in 
coincidence.  If  not,  loosen  the  screws  that  attach  the 
vernier  to  the  standards,  and,  moving  it  so  as  to  bring 
its  zero  half-way  to  the  vertical  circle's  zero,  make  it 
secure  again.  Repeat  until  no  error  can  be  detected. 
Instead  of  moving  the  vernier  zero,  the  circle  zero  may 
be  brought  into  coincidence  with  it,  and  the  horizontal 
wire  moved  back  over  half  the  amount  by  which  it  has 
been  thus  displaced,  provided  the  error  is  so  slight  as 
not  to  appreciably  throw  the  intersection  of  the  cross- 
wires  out  of  the  center  of  the  field  of  view  as  previously 
adjusted.  Repeat  as  before. 

6.  a.  Level  on  Telescope.  —  Level  the  instrument  care- 
fully, and  with  the  clamp-aud-tangent  movement  to  the 
horizontal  axis,  bring  the  zero  of  the  vertical  circle  into 
coincidence  with  the  vernier  zero,  and,  by  means  of  the 
screws  at  each  end  of  the  level,  bring  its  bubble  to  the 
center,  taking  care  not  to  jar  the  instrument  out  of 
level. 

5.  b.  Level  on  Telescope.  —  (When  the  instrument 
has  only  a  vertical  arc  and  not  a  full  circle.)  On  toler- 
ably level  ground  stake  four  points  in  line  and  equi- 
distant (about  100  feet),  calling  them  (say)  jD,  «7i,  E,  J~3, 
consecutively.  (It  is  well  that  D  should  not  be  lower 
than  E,  nor  J:]  than  Jj.)  Set  up  the  instrument  at  Ji, 
and  direct  the  line  of  sight  to  a  graduated  rod  held,  as 


40  METHODS   OF   ADJUSTMENT. 

nearly  as  may  be,  vertical  on  the  higher  of  stakes  Z),  E 
(say  D),  taking  a  small  reading  d-i.  Clamp  the  telescope, 
and  turning  the  instrument  around,  take  the  reading  et 
on  E.  Set  up  and  level  the  instrument  at  J~3,  so  that 
the  height  of  the  telescope  shall  be  greater  than  that 
corresponding  to  the  readings  d1?  e^.  Unclamp  the  tele- 
scope and  direct  the  line  of  sight  so  that,  without  changing 
its  position,  readings  e2,  d2,  respectively  greater  than 
e1?  dj,  are  obtained  on  the  rod  held  successively  on  E 
and  D.  From  three  times  e2  —  e\  take  d2  —  di  and  divide 
the  result  by  2.  Set  the  target  at  a  reading  greater  by 
this  calculated  amount  than  d1?  and,  holding  the  rod 
on  7),  bisect  the  target  by  the  line  of  sight  and  clamp 
the  telescope.  By  means  of  the  screws  at  the  end  of 
the  level,  bring  its  bubble  to  the  center,  and  see  if  the 
line  of  sight  still  bisects  the  target.  If  it  has  been 
jarred  out  of  position,  put  it  back  and  again  bring  the 
bubble  to  the  center.  When  the  line  of  sight  bisects 
the  target  and  the  bubble  is  in  the  center,  the  adjustment 
is  complete. 

A  simplification  of  the  proceeding  is  as  follows : 
Instead  of  taking  the  first  two  readings,  drive  stakes  so 
that  their  tops  are  cut  by  the  line  of  sight,  the  telescope 
being  clamped.  Their  tops  are  then  in  the  same  hori- 
zontal line.  Making  the  line  of  sight  as  nearly  horizontal 
as  possible  by  estimation,  take  a  reading  on  the  nearer 
of  these  two  stakes,  and  then,  holding  the  rod  on  the 
farther  one  (target  at  same  reading) ,  if  the  line  of  sight 


METHODS    OF    ADJUSTMENT.  41 

does  not  bisect  the  target,  turn  the  telescope  by  the 
tangent-screw  so  that  it  will ;  repeating  this  until  it  will 
bisect  the  target  held  on  the  far  stake  at  the  same  reading 
as  on  the  near  one.  Then  bring  the  bubble  to  the  center 
by  the  screws  at  the  ends  of  the  level- tube  as  before. 

6.  b.  The  Vertical  Arc.  —With  the  bubble  of  the  level 
on  the  telescope  at  the  center,  clamp  the  vertical  arc  to  the 
axis  of  the  telescope,  with  its  zero  in  coincidence  with 
that  of  the  vernier,  and  it  is  in  condition  to  correctly 
measure  vertical  angles. 

III.   THE  LEVEL. 

1.  The  Line  of  Collimation.  —  Set  the  tripod  firmly, 
remove  the  wye-pins  from  the  clips,  so  that  the  telescope 
may  be  turned  in  its  bearings,  and,  by  means  of  the 
leveling  and  tangent  screws,  bring  either  of  the  wires 
into  coincidence  with  a  clearly  marked  edge  of  some 
object,  from  two  to  five  hundred  feet  distant.  Then 
turn  the  telescope  half-way  around  in  the  w}'es,  so  that 
the  same  wire  may  be  compared  with  the  edge  selected. 
If  it  now  coincides  with  the  edge,  it  is  in  adjustment ; 
if  not,  bring  it  half-way  to  it  by  moving  the  capstan- 
head  screws  at  right  angles  to  the  wire  in  question, 
remembering  the  inverting  property  of  the  eye-piece. 
Repeat  until  it  will  reverse  correctly.  Then  adjust  the 
other  wire  in  the  same  manner ;  or,  if  their  errors  are 


42  METHODS   OF  ADJUSTMENT. 

great,  make  them  nearly  correct  before  exactly  adjusting 
either. 

When  this  has  been  effected,  unscrew  the  covering  of 
the  eye-piece  centering-screws,  and  move  each  pair  in 
succession  so  as  to  bring  the  center  of  the  field  of  view 
to  coincide  with  the  intersection  of  the  cross- wires,  test- 
ing the  centering  by  revolving  the  telescope  in  the  wyes, 
when  the  object  should  not  appear  to  move.  The  screws 
in  this  case  are  to  be  moved  as  it  appears  they  should 
be.  Replace  the  covering  of  the  screws. 

2,  The  Level-Bubble.  —  Having  the  plate  about  hori- 
zontal, place  the  telescope  over  either  pair  of  leyeling- 
screws,  and,  clamping  the  instrument,  remove  the  wye- 
pins  and  bring  the  bubble  to  the  center  by  means  of  this 
pair  of  leveling-screws.  Reverse  the  telescope  end  for 
end  as  to  the  wyes,  replacing  it  with  the  level-tube 
immediately  beneath,  and  note  whether  the  bubble 
remains  at  the  center.  Now  rotate  the  telescope  slightly 
to  each  side,  and  see  if  this  causes  the  bubble  to  move 
toward  either  end.  If,  after  the  reversal  it  was  at  the 
center,  and  remained  so  during  the  slight  rotation,  the 
level-bubble  is  in  adjustment. 

If  the  rotation  causes  it  to  move  from  its  position  after 
reversal  toward  either  end,  adjust,  by  trial,  the  horizontal 
screw  so  that  this  will  not  be  the  case.  Repeat  this 
reversa[,  etc.,  until  no  further  adjustment  of  the  horizontal 
screw  is  needed.  Then  (the  telescope  being  in  the 


METHODS   OF  ADJUSTMENT.  43 

second  position)  bring  the  bubble  half-way  to  the  center 
by  means  of  the  vertical  screw-nuts  at  one  end  of  the 
level  tube-case,  and  repeat  the  whole  process  until  the 
bubble  will  remain  at  tlie  center  after  reversal  and  slight 
rotation. 

3.  The  Wyes.  —  Having  the  telescope  in  its  normal 
position  as  to  the  wyes,  place  it  over  a  pair  of  the  level- 
ing-screws  and  bring  the  bubble  to  the  center  by  means 
of  them.  Turn  the  instrument  half-way  around  on  the 
spindle,  and,  if  the  bubble  runs  toward  either  end,  bring 
it  half-way  back  by  the  wye-nuts  on  either  end  of  the 
bar  and  the  rest  of  the  way  by  the  pair  of  leveling-screws. 
Then  place  the  telescope  over  the  other  pair,  and  proceed 
in  the  same  way,  changing  to  each  pair  successively 
until  the  adjustment  is  completely  effected  for  each,  so 
that  the  bubble  will  remain  at  the  center  during  an  entire 
revolution  of  the  telescope  on  the  spindle. 


188 


MATHEMATICS. 


Peirce's  Three  and  Four  Place  Tables  of  Loga- 

rithinic  and  Trigonometric  Functions.  By  JAMES  MILLS  PEIRCE, 
University  Professur  of  Mathematics  in  Harvard  University.  Quarto. 
Cloth.  Mailing  Price,  45  cts.  j  Introduction,  40  cts. 

Four-place  tables  require,  in  the  long  ran,  only  half  as  much  time 
.s  five-place  tables,  one-third  as  much  time  as  six-place  tables,  and 
one-fourth  as  much  as  those  of  seven  places.  They  are  sufficient 
for  the  ordinary  calculations  of  Surveying,  Civil,  Mechanical,  and 
Mining  Engineering,  and  Navigation ;  for  the  work  of  the  Physical 
or  Chemical  Laboratory,  and  even  for  many  computations  of  Astron- 
omy. They  are  also  especially  suited  to  be  used  in  teaching,  as  they 
illustrate  principles  as  well  as  the  larger  tables,  and  with  far  less 
expenditure  of  time.  The  present  compilation  has  been  prepared 
with  care,  and  is  handsomely  and  clearly  printed. 


Elements  of  the  Differential  Calculus. 

With  Numerous  Examples  and  Applications.  Designed  for  Use  as  a 
College  Text-Book.  By  W.  E.  BYEKLY,  Professor  of  Mathematics, 
Harvard  University.  8vo.  273  pages.  Mailing  Price,  $2.15  ;  Intro- 
duction, $2.00. 

This  book  embodies  the  results  of  the  author's  experience  in 
teaching  the  Calculus  at  Cornell  and  Harvard  Universities,  and  is 
intended  for  a  text-book,  and  not  for  an  exhaustive  treatise.  Its 
peculiarities  are  the  rigorous  use  of  the  Doctrine  of  Limits,  as  a 
foundation  of  the  subject,  and  as  preliminary  to  the  adoption  of  the 
more  direct  and  practically  convenient  infinitesimal  notation  and 
nomenclature  ;  the  early  introduction  of  a  few  simple  formulas  and 
methods  for  integrating;  a  rather  elaborate  treatment  of  the  use  of 
infinitesimals  in  pure  geometry ;  and  the  attempt  to  excite  and  keep 
up  the  interest  of  the  student  by  bringing  in  throughout  the  whole 
book,  and  not  merely  at  the  end,  numerous  applications  to  practical 
problems  in  geometry  and  mechanics. 


James  Mills  Peirce,  Prof,  of 
Math.,  Harvard  Univ.  (From  the  Har- 
vard Register}  /In  mathematics,  as  in 
other  branches  of  study,  the  need  is 
now  very  much  felt  of  teaching  which 


is  general  without  being  superficial; 
limited  to  leading  topics,  and  yet  with- 
in its  limits;  thorough,  accurate,  and 
practical;  adapted  to  the  communica- 
tion of  some  degree  of  power,  as  well 


MA  THEM  A  TICS. 


189 


as  knowledge,  but  free  from  details 
which  are  important  only  to  the  spe- 
cialist. Professor  Byerly's  Calculus 
appears  to  be  designed  to  meet  this 
want.  .  .  .  Such  a  plan  leaves  much 
room  for  the  exercise  of  individual 
judgment;  and  differences  of  opinion 
will  undoubtedly  exist  in  regard  to  one 
and  another  point  of  this  book.  But 
all  teachers  will  agree  that  in  selection, 
arrangement,  and  treatment^,  it  is,  on 
the  whole,  in  a  very  high  degree,  wise, 
able,  marked  by  a  true  scientific  spirit, 
and  calculated  to  develop  the  same 
spirit  in  the  learner.  .  .  .  The  book 
contains,  perhaps,  all  of  the  integral 
calculus,  as  well  as  of  the  differential, 
that  is  necessary  to  the  ordinary  stu- 
dent. And  with  so  much  of  this  great 
scientific  method,  every  thorough  stu- 
dent of  physics,  and  every  general 
scholar  who  feels  any  interest  in  the 
relations  of  abstract  thought,  and  is 
capable  of  grasping  a  mathematical 
idea,  ought  to  be  familiar.  One  who 
aspires  to  technical  learning  must  sup- 
plement his  mastery  of  the  elements 
by  the  study  of  the  comprehensive 
theoretical  treatises.  .  .  .  But  he  who  is 
thoroughly  acquainted  with  the  book 
before  us  has  made  a  long  stride  into 
a  sound  and  practical  knowledge  of 
the  subject  of  the  calculus.  He  has 
begun  to  be  a  real  analyst. 

H.  A.  Newton,  Prof,  of  Math,  in 
Yale  Coll.,  New  Haven  :  I  have  looked 
it  through  with  care,  and  find  the  sub- 
ject very  clearly  and  logically  devel- 
oped. I  a/n  strongly  inclined  to  use  it 
in  my  class  next  year. 

S.  Hart-,  recent  Prof,  of  Math,  in 
Trinity  Coll.,  Conn. :  The  student  can 
hardly  fail,  I  think,  to  get  from  the  book 
an  exact,  and,  at  the  same  time,  a  satis- 
factory explanation  of  the  principles  on 
which  the  Calculus  is  based;  and  the 
introduction  of  the  simpler  methods  of 


integration,  as  they  are  needed,  enables 
applications  of  those  principles  to  be 
introduced  in  such  a  way  as  to  be  both 
interesting  and  instructive. 

Charles  Kraus,  Techniker,  Pard- 
tibitz,  Bohemia,  Austria  :  Indem  ich 
den  Empfang  Ihres  Buches  dankend 
bestaetige  muss  ich  Ihnen,  hoch  geehr- 
ter  Herr  gestehen,  dass  mich  dasselbe 
sehr  erfreut  hat,  da  es  sich  durch 
grosse  Reichhaltigkeit,  besonders  klare 
Schreibvveise  und  vorzuegliche  Behand- 
lung  des  Stoffes  auszeichnet,  und  er- 
vveist  sich  dieses  Werk  als  eine  bedeut- 
ende  Bereicherung  der  mathematischen 
Wissenschaft. 

De  Volson  Wood,  Prof,  of 
Math.,  Stevens'  Inst.,  Hoboken,  N.J.: 
To  say,  as  I  do,  that  it  is  a  first-class 
work,  is  probably  repeating  what  many 
have  already  said  for  it.  I  admire  the 
rigid  logical  character  of  the  work, 
and  am  gratified  to  see  that  so  able  a 
writer  has  shown  explicitly  the  relation 
between  Derivatives,  Infinitesimals,  and 
Differentials.  The  method  of  Limits 
is  the  true  one  on  which  to  found  the 
science  of  the  calculus.  The  work  is 
not  only  comprehensive,  but  no  vague- 
ness is  allowed  in  regard  to  definitions 
or  fundamental  principles. 

Del  Kemper,  Prof,  of  Math., 
Hanipden  Sidney  Coll.,  Va. :  My  high 
estimate  of  it  has  been  amply  vindi- 
cated by  its  use  in  the  class-room. 

R.  H.  Graves,  Prof,  of  Math., 
Univ.  of  North  Carolina  :  I  have  al- 
ready decided  to  use  it  with  my  next 
class ;  it  suits  my  purpose  better  than 
any  other  book  on  the  same  subject 
with  which  I  am  acquainted. 

Edw.  Brooks,  Author  of  a  Series 
of  Math.  :  Its  statements  are  clear  and 
scholarly,  and  its  methods  thoroughly 
analytic  and  in  the  spirit  of  the  latest 
mathematical  thought. 


190 


MA  THE  MA  TICS. 


Syllabus  of  a  Course  in  Plane  Trigonometry. 

By  W.  E.  BYERLY.     8vo.     8  pages.     Mailing  Price,  10  cts. 

Syllabus  of  a  Course  in  Plane  Analytical  Geom- 

etry.     By  W.  E.  BYERLY.     8vo.     12  pages.     Mailing  Price,  10  cts. 

Syllabus  of  a  Course  in  Plane  Analytic  Geom- 

etry     (Advanced  Course.}     By  W.   E.  BYERLY,  Professor  of  Mathe- 
matics, Harvard  University.     8vo.      12  pages.     Mailing  Price,  10  cts. 

Syllabus  of  a  Course  in  Analytical  Geometry  of 


Three  Dimensions.     By  W.  E.  BYERLY. 
Price,  10  cts. 


8vo.     10  pages.      Mailing 


Syllabus  of  a  Course  on  Modern  Methods   in 

Analytic  Geometry.     By  W.  E.  BYERLY.      8vo.      8  pages.      Mailing 
Price,  10  cts. 

Syllabus  of  a  Course  in  the  Theory  of  Equations. 

By  W.  E.  BYERLY.     8vo.     8  pages.     Mailing  Price,  10  cts. 

Elements  of  the  Integral  Calculus. 

By  W.  E.  BYERLY,  Professor  of  Mathematics  in  Harvard  University. 
8vo.     204  pages.     Mailing  Price,  #2.15;  Introduction,  $2.00. 

This  volume  is  a  sequel  to  the  author's  treatise  on  the  Differential 
Calculus  (see  page  134),  and,  like  that,  is  written  as  a  text-book. 
The  last  chapter,  however,  —  a  Key  to  the  Solution  of  Differential 
Equations,  —  may  prove  of  service  to  working  mathematicians. 


H.  A.  Newton,  Prof,  of  Math., 
Yale  Coll. :  We  shall  use  it  in  my 
optional  class  next  term. 

Mathematical  Visitor :  The 
subject  is  presented  very  clearly.  It  is 
the  first  American  treatise  on  the  Cal- 
culus that  we  have  seen  which  devotes 
any  space  to  average  and  probability. 

Schoolmaster,  London :  The 
merits  of  this  work  are  as  marked  as 


those  of  the  Differential   Calculus  by 
the  same  author. 

Zion's  Herald  :  A  text- book  every 
way  worthy  of  the  venerable  University 
in  which  the  author  is  an  honored 
teacher.  Cambridge  in  Massachusetts, 
like  Cambridge  in  England,  preserves 
its  reputation  for  the  breadth  and  strict- 
ness of  its  mathematical  requisitions, 
and  these  form  the  spinal  column  of  a 
liberal  education. 


MA  Til  EM  A  TICS. 


191 


A  Short  Table  of  Integrals. 


To  accompany  BYERLY'S  INTEGRAL  CALCULUS.  By  B.  O. 
PEIRCE,  JR.,  Instructor  in  Mathematics,  Harvard  University.  16  pages. 
Mailing  Price,  10  cts.  To  be  bound  with  future  editions  of  the  Calculus. 


Elements  of  Quaternions. 

By  A.  S.  HARDY,  Ph.D.,  Professor  of  Mathematics,  Dartmouth  College. 
Crown  8vo.  Cloth.  240  pages.  Mailing  Price,  $2.15;  Introduction, 
$2.00. 

The  chief  aim  has  been  to  meet  the  wants  of  beginners  in  the 
class-room.  The  Elements  and  Lectures  of  Sir  W.  R.  Hamilton 
are  mines  of  wealth,  and  may  be  said  to  contain  the  suggestion 
of  all  that  will  be  done  in  the  way  of  Quaternion  research  and 
application :  for  this  reason,  as  also  on  account  of  their  diffuseness 
of  style,  they  are  not  suitable  for  the  purposes  of  elementary  instruc- 
tion. The  same  may  be  said  of  Tait's  Q^^aternions,  a  work  of 
great  originality  and  comprehensiveness,  in  style  very  elegant  but 
very  concise,  and  so  beyond  the  time  and  needs  of  the  beginner. 
The  Introduction  to  Quaternions  by  Kelland  contains  many  exer- 
cises and  examples,  of  which  free  use  has  been  made,  admirably 
illustrating  the  Quaternion  spirit  and  method,  but  has  been  found, 
in  the  class-room,  practically  deficient  in  the  explanation  of  the 
theory  and  conceptions  which  underlie  these  applications.  The 
object  in  view  has  thus  been  to  cover  the  introductory  ground  more 
thoroughly,  especially  in  symbolic  transformations,  and  at  the  same 
time  to  obtain  an  arrangement  better  adapted  to  the  methods  of 
instruction  common  in  this  country. 


FROM   COLLEGE   PROFESSORS. 


James  Mills  Peirce,  Prof,  of 
Math.,  Harvard  Coll. :  1  am  much 
pleased  with  it.  It  seems  to  me  to 
supply  in  a  very  satisfactory  manner 
the  need  which  has  long  existed  of  a 
clear,  concise,  well-arranged,  and  logi- 
cally-developed introduction  to  this 
branch  of  Mathematics.  I  think  Prof. 
Hardy  has  shown  excellent  judgment 
in  his  methods  of  treatment,  and  also 
in  limiting  himself  to  the  exposition 


and  illustration  of  the  fundamental 
principles  of  his  subject.  It  is,  as  it 
ought  to  be,  simply  a  preparation  for 
the  study  of  the  writings  of  Hamilton 
and  Tail. 

Charles  A.  Young,  Prof,  of 
Astronomy,  Princeton  Coll.  :  I  find  it 
by  far  the  most  clear  and  intelligible 
statement  of  the  matter  I  have  yet 
seen. 


192 


MA  THEM  A  TICS. 


Elements  of  the  Differential  and  Integral  Calculus. 

With  Examples  and  Applications.  By  J.  M.  TAYLOR,  Professor  c.f 
Mathematics  in  Madison  University.  8vo.  Cloth.  249  pp.  Mailing 
price,  $1.95;  Introduction  price,  $i.So. 

The  aim  of  this  treatise  is  to  present  simply  and  concisely  the 
fundamental  problems  of  the  Calculus,  their  solution,  and  more 
common  applications.  Its  axiomatic  datum  is  that  the  change  of  a 
variable,  when  not  uniform,  may  be  conceived  as  becoming  uniform 
at  any  value  of  the  variable. 

It  employs  the  conception  of  rates,  which  affords  finite  differen- 
tials, and  also  the  simplest  and  most  natural  view  of  the  problem  of 
the  Differential  Calculus.  This  problem  of  finding  the  relative 
rates  of  change  of  related  variables  is  afterwards  reduced  to  that  of 
finding  the  limit  of  the  ratio  of  their  simultaneous  increments  ;  and, 
in  a  final  chapter,  the  latter  problem  is  solved  by  the  principles  of 
infinitesimals. 

Many  theorems  are  proved  both  by  the  method  of  rates  and  that 
of  limits,  and  thus  each  is  made  to  throw  light  upon  the  other. 
The  chapter  on  differentiation  is  followed  by  one  on  direct  integra- 
tion and  its  more  important  applications.  Throughout  the  work 
there  are  numerous  practical  problems  in  Geometry  and  Mechanics, 
which  serve  to  exhibit  the  power  and  use  of  the  science,  and  to 
excite  and  keep  alive  the  interest  of  the  student. 

Judging  from  the  author's  experience  in  teaching  the  subject,  it 
is  believed  that  this  elementary  treatise  so  sets  forth  and  illustrates 
the  highly  practical  nature  of  the  Calculus,  as  to  awaken  a  lively 
interest  in  many  readers  to  whom  a  more  abstract  method  of  treat- 
ment would  be  distasteful. 


Oren  Root,  Jr.,  Prof,  of  Afath., 
Hamilton  Coll.,  N.Y.:  In  reading  the 
manuscript  I  was  impressed  by  the 
clearness  of  definition  and  demonstra- 
tion, the  pertinence  of  illustration,  and 
the  happy  union  of  exclusion  and  con- 
densation. It  seems  to  me  most  admir- 
ably suited  for  use  in  college  classes. 
I  prove  my  regard  by  adopting  this  as 
our  text-book  on  the  calculus. 


C.    M.    Charrappin,    8.J.,   St. 

Louis  Univ.  :  I  have  given  the  book  a 
thorough  examination,  and  I  am  satis- 
fied that  it  is  the  best  work  on  the  sub- 
ject I  have  seen.  I  mean  the  best 
work  for  what  it  was  intended, — a  text- 
book. I  would  like  very  much  to  in- 
troduce it  in  the  University. 
(Jan.  12,  1885.) 


MATHEMATICS. 


193 


J.  G.  Fox,  Prof,  of  Civil  Eng.,  La- 
fayette Coll.,  Easton,  Pa.:  It  has  some 
very  good  points  in  its  favor,  such  as, 
the  arrangement  of  the  subject-matter, 
the  "  numerous  practical  problems," 
etc.  (Feb.  21,  1885.) 

J.  Howard  Gore,  Prof,  of  Math., 

Columbian  Coll.,  Washington,  D.C.  : 
From  a  careful  inspection  I  think  that 
in  very  many  respects  it  is  a  marked 
improvement  on  the  various  works  on 
calculus  now  in  use.  I  have  always 
thought  that  integral  and  differential 
calculus  should  be  studied  at  the  same 
time.  This  is  not  feasible  except  when 
the  author  arranged  the  subject-matter 
with  that  plan  in  view,  as  in  this  book. 
At  present  I  regard  with  favor  the 
introduction  of  this  work  in  my  classes 
next  session.  (Jan.  9,  1885.) 

C.  H.  Judson,  Prof,  of  Math., 
Fur  man  Univ.,  Greenville,  S.C.:  I  find 
it  to  be  one  of  the  most  accurate,  logical, 
and  carefully  prepared  text-books  that 
I  have  met  with.  I  believe  there  is  no 
better  text-book  for  teachers  who  adopt 
the  theory  of  "  Rates  "  as  the  basis  of 
the  calculus.  (Dec.  30,  1884.) 

O.  C.  Gray,  Prof,  of  Math.,  Ark. 
Indus.  Univ. :  Thus  far,  I  am  very 
much  pleased  with  it,  particularly  in 
the  fact  that  chapters  on  Differentiation 
and  Integration  immediately  follow 
each  other.  Such  an  arrangement  was 
needed.  (Jan.  6,  1885.) 

P.  L.  Morse,  Prof,  of  Math.. 
Hanover  Coll.,  Ind. :  The  matter  is 
certainly  admirably  chosen,  and  the 
arrangement  natural  and  unique.  The 
Integral  Calculus  is  placed  in  proper 
order,  and  the  practical  application  to 
Mechanics  will  do  much  to  clear  away 
the  mysteries  of  which  the  student  often 
justly  complains.  (Dec.  25,  1884.) 

London  Schoolmaster:  This 
easy  but  comprehensive  treatise  on  the 


calculus  differs  in  its  methods  from 
similar  text-books  produced  on  this 
side  of  the  Atlantic.  Altogether,  con- 
sidering the  extent  of  the  ground  it 
covers,  it  is  one  of  the  easiest  and 
clearest  text-books  on  the  calculus 
we  know. 

Boston  Advertiser :  It  reflects  a 
high  measure  of  credit  upon  the  au- 
thor. He  certainly  has  succeeded  to  a 
degree  that  may  well  insure  for  the 
present  volume  extended  use  as  a  text- 
book in  our  colleges.  He  has  shown  a 
thorough,  comprehensive  grasp  of  his 
subject,  and  has  brought  this  to  bear 
with  singular  force  in  his  pointed  defi- 
nitions, and  in  the  clear  reasoning  of 
his  demonstrations. 

The  Nation,  New  York:  It  has 
two  marked  characteristics.  In  the 
first  place,  it  is  evidently  a  most  care- 
fully written  book.  There  is  nothing 
vague  or  slipshod  in  it.  Nearly  every 
sentence,  certainly  every  theorem,  seems 
I  to  have  been  constructed  with  a  stren- 
uous effort  to  give  it  clearness  and  pre- 
cision. This  constant  attention  to  the 
form  of  expression  has  enabled  the 
author  to  be  concise  without  becoming 
obscure.  We  are  acquainted  with  no 
text-book  of  the  calculus  which  com- 
presses so  much  matter  into  so  few 
pages,  and  at  the  same  time  leaves  the 
impression  that  all  that  is  necessary 
has  been  said.  In  the  second  place, 
the  number  of  carefully  selected  ex- 
amples, both  of  those  worked  out  in 
full  in  illustration  of  the  text,  and  of 
those  left  for  the  student  to  work  out 
for  himself,  is  extraordinary.  From 
this  point  of  view  those  teachers  and 
pupils  who  are  accustomed  to  or  prefer 
a  different  text-book,  would  still  do 
well  to  provide  themselves  with  this, 
regarding  it  merely  as  a  collection  of 
examples  and  without  any  reference  to 
the  text. 


194  MA  THEM  A  TICS. 


Metrical  Geometry :  An  Elementary  Treatise  on 

Mensuration.  By  GEORGE  BRUCE  HALSTED,  Ph.D.,  Prof.  Mathema- 
tics, University  of  Texas,  Austin.  I2mo.  Cloth.  246  pages.  Mailing 
price,  $1.10;  Introduction,  $1.00. 

This  work  applies  new  principles  and  methods  to  simplify  the 
measurement  of  lengths,  angles,  areas,  and  volumes.  It  is  strictly 
demonstrative,  but  uses  no  Trigonometry,  and  is  adapted  to  be  taken 
up  in  connection  with,  or  following  any  elementary  Geometry.  It 
treats  of  accessible  and  inaccessible  straight  lines,  and  of  their  inter- 
dependence when  in  triangles,  circles,  etc. ;  also  gives  a  more  rigid 
rectification  of  the  circumference,  etc.  It  introduces  the  natural 
unit  of  angle,  and  deduces  the  ordinary  and  circular  measure. 
Enlisting  the  auxiliary  powers  which  modern  geometers  have  recog- 
nized in  notation,  it  binds  up  each  theorem  also  in  a  self-explanatory 
formula,  and  this  throughout  the  whole  book  on  a  system  which 
renders  confusion  impossible,  and  surprisingly  facilitates  acquire- 
ment, as  has  been  tested  with  very  large  classes  in  Princeton  College. 
In  addition  to  all  the  common  propositions  about  areas,  a  new 
method,  applicable  to  any  polygon,  is  introduced,  which  so  simplifies 
and  shortens  all  calculations,  that  it  is  destined  to  be  universally 
adopted  in  surveying,  etc.  In  addition  to  the  circle,  sector,  segment, 
zone,  annulus,  etc.,  the  parabola  and  ellipse  are  measured ;  and  be- 
sides the  common  broken  and  curved  surfaces,  the  theorems  of 
Pappus  are  demonstrated.  Especial  mention  should  be  made  of  the 
treatment  of  solid  angles,  which  is  original,  introducing  for  the  first 
time,  we  think,  the  natural  unit  of  solid  angle,  and  making  spherics 
easy.  For  solids,  a  single  informing  idea  is  fixed  upon  of  such 
fecundity  as  to  place  within  the  reach  of  children  results  heretofore 
only  given  by  Integral  Calculus.  Throughout,  a  hundred  illustrative 
examples  are  worked  out,  and  at  the  end  are  five  hundred  carefully 
arranged  and  indexed  exercises,  using  the  metric  system. 

OPINIONS. 


Simon  Newcomb,  Nautical  Al- 
manac Office,  Washington,  D.C.:  I  am 
much  interested  in  your  Mensuration, 
and  wish  I  had  seen  it  in  time  to  have 
sonje  exercises  suggested  by  it  put  into 
my  Geometry.  (Sept.  8,  1881.) 


Alexander  MacParlane,  Exam- 
iner in  Mathematics  to  the  University 
of  Edinburgh,  Scotland :  The  method, 
figures,  and  examples  appear  excellent, 
and  I  anticipate  much  benefit  from  its 
minute  perusal. 


MA  THE  MA  TICS.  1 95 


Elementary  Co-ordinate  Geometry. 

By  W.  B.  SMITH,  Professor  of  Physics,  Missouri  State  University.     I2mo. 
Cloth.     312  pp.     Mailing  price,  $2.15;   for  Introduction,  $2.00. 

While  in  the  study  of  Analytic  Geometry  either  gain  of  knowledge 
or  culture  of  mind  may  be  sought,  the  latter  object  alone  can  justify 
placing  it  in  a  college  curriculum.  Yet  the  subject  may  be  so  pur- 
sued as  to  be  of  no  great  educational  value.  Mere  calculation,  or  the 
solution  of  problems  by  algebraic  processes,  is  a  very  inferior  dis- 
cipline of  reason.  Even  geometry  is  not  the  best  discipline.  In  all 
thinking  the  real  difficulty  lies  in  forming  clear  notions  of  things. 
In  doing  this  all  the  higher  faculties  are  brought  into  play.  It  is  this 
formation  of  concepts,  therefore,  that  is  the  essential  part  of  mental 
training.  He  who  forms  them  clearly  and  accurately  may  be  safely 
trusted  to  put  them  together  correctly.  Nearly  every  seeming  mis- 
take in  reasoning  is  really  a  mistake  in  conception. 

Such  considerations  have  guided  the  composition  of  this  book. 
Concepts  have  been  introduced  in  abundance,  and  the  proofs  made 
to  hinge  directly  upon  them.  Treated  in  this  way  the  subject 
seems  adapted,  as  hardly  any  other,  to  develop  the  power  of 
thought. 

Some  of  the  special  features  of  the  work  are :  — 

1.  Its   SIZE   is   such   it   can   be   mastered  in  the  time  generally 
allowed. 

2.  The  SCOPE  is  far  wider  than  in  any  other  American  work. 

3.  The  combination  of  small  size  and  large  scope  has  been  secured 
through  SUPERIOR  METHODS,  —  modern,  direct,  and  rapid. 

4.  Conspicuous  among  such  methods  is  that  of  DETERMINANTS, 
here  presented,  by  the  union  of  theory  and  practice,  in  its  real 
power  and  beauty. 

5.  Confusion   is  shut  out  by  a  consistent,  and    self-explaining 

NOTATION. 

6.  The  ORDER  OF  DEVELOPMENT  is  natural,  and   leads  without 
break  or  turn  from  the  simplest  to  the  most  complex.     The  method 
is  heuristic. 

7.  The  student's  grasp  is  strengthened  by  numerous  EXERCISES. 

8.  The  work  has  been  TESTED  at  every  point  IN  THE  CLASS- 
ROOM. 


196  MA  THEM  A  TICS. 


Determinants. 

The  Theory  of  Determinants :  an  Elementary  Treatise.  By  PAUL  H. 
HANUS,  B.S.,  Professor  of  Mathematics  in  the  University  of  Colorado. 
8vo.  Cloth,  ooo  pages.  Mailing  price,  $0.00;  for  Introduction,  $0.00. 

This  is  a  text-book  for  the  use  of  students  in  colleges  and  tech- 
nical schools.  The  need  of  an  American  work  on  determinants 
has  long  been  felt  by  all  teachers  and  students  who  have  extended 
their  reading  beyond  the  elements  of  mathematics.  The  importance 
of  the  subject  is  no  longer  overlooked.  The  shortness  and  elegance 
imparted  to  many  otherwise  tedious  processes,  by  the  introduction 
of  determinants,  recommend  their  use  even  in  the  more  elementary 
branches,  while  the  advanced  student  cannot  dispense  with  a  knowl- 
edge of  these  valuable  instruments  of  research.  Moreover,  deter- 
minants are  employed  by  all  modern  writers. 

This  book  is  written  especially  for  those  who  have  had  no  previous 
knowledge  of  the  subject,  and  is  therefore  adapted  to  self-instruction 
as  well  as  to  the  needs  of  the  class-room.  To  this  end  the  subject 
is  at  first  presented  in  a  very  simple  manner.  As  the  reader  ad- 
vances, less  and  less  attention  is  given  to  details.  Throughout  the 
entire  work  it  is  the  constant  aim  to  arouse  and  enliven  the  reader's 
interest  by  first  showing  how  the  various  concepts  have  arisen 
naturally,  and  by  giving  such  applications  as  can  be  presented  with- 
out exceeding  the  limits  of  the  treatise.  The  work  is  sufficiently 
comprehensive  to  enable  the  student  that  has  mastered  the  volume 
to  use  the  determinant  notation  with  ease,  and  to  pursue  his  further 
reading  in  the  modern  higher  algebra  with  pleasure  and  profit. 

In  Chapter  I.  the  evolution  of  a  theory  of  determinants  is  touched 
upon,  and  it  is  shown  how  determinants  are  produced  in  the  process 
of  eliminating  the  variables  from  systems  of  simple  equations  with 
some  further  preliminary  notions  and  definitions. 

In  Chapter  II.  the  most  important  properties  of  determinants  are 
discussed.  Numerous  examples  serve  to  fix  and  exemplify  the  prin- 
ciples deduced. 

Chapter  III.  comprises  half  the  entire  volume.  It  is  the  design 
of  this  chapter  to  familiarize  the  reader  with  the  most  important 
special  forms  that  occur  in  application,  and  to  enable  him  to  realize 
the  practical  usefulness  of  determinants  as  instruments  of  research. 

[Ready  June  i. 


MATHEMATICS.  197 


Examples  of  Differential  Equations. 

By  GEORGE  A.  OSBORNE,  Professor  of  Mathematics  in  the  Massachusetts 
Institute  of  Technology,  Boston.  I2mo.  Cloth,  viii  -f  50  pp.  Mail- 
ing price,  60  cts.;  for  Introduction,  50  cts. 

Notwithstanding  the  importance  of  the  study  of  Differential  Equa- 
tions, either  as  a  branch  of  pure  mathematics,  or  as  applied  to 
Geometry  or  Physics,  no  American  work  on  this  subject  has  been 
published  containing  a  classified  series  of  examples.  This  book  is 
intended  to  supply  this  want,  and  provides  a  series  of  nearly  three 
hundred  examples  with  answers  systematically  arranged  and  grouped 
under  the  different  cases,  and  accompanied  by  concise  rules  for  the 
solution  of  each  case. 

It  is  hoped  that  the  work  will  be  found  useful,  not  only  in  the 
study  of  this  important  subject,  but  also  by  way  of  reference  to 
mathematical  students  generally  whenever  the  solution  of  a  differen- 
tial equation  is  required. 

Elements  of  the  Theory  of  the  Newtonian  Poten- 

tl'al  Function.  By  B.  O.  PEIRCE,  Assistant  Professor  of  Mathematics 
and  Physics,  Harvard  University.  I2mo.  Cloth.  154  pages.  Mailing 
price,  $l.6o;  for  Introduction,  $1.50. 

A  knowledge  of  the  properties  of  this  function  is  essential  for 
electrical  engineers,  for  students  of  mathematical  physics,  and  for 
all  who  desire  more  than  an  elementary  knowledge  of  experimental 
physics. 

This  book,  based  upon  notes  made  for  class-room  use,  was  written 
because  no  book  in  English  gave  in  simple  form,  for  the  use  of 
students  who  know  something  of  the  calculus,  so  much  of  the  theory 
of  the  potential  function  as  is  needed  in  the  study  of  physics. 
Both  matter  and  arrangement  have  been  practically  adapted  to  the 
end  in  view. 

CHAPTER  I.  The  Attraction  of  Gravitation. 

II.  The  Newtonian  Potential  Function  in  the  Case  of  Gravitation. 

III.  The  Newtonian  Potential  Function  in  the  Case  of  Repulsive 

Forces. 

IV.  Surface  Distributions.     Green's  Theorem. 

V.  Application    of  the    Results    of  the    Preceding  Chapters   to 
Electrostatics. 


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